Elementary Theory in Linearly Tapered Beams

With the use of a generalized Kirchhoff hypothesis in which the transverse shear strain in cylindrical coordinates is assumed to be zero, a beam theory is developed for linearly tapered members. The equations are analogous to those of conventional beam theory but they are applicable to a different class of problems. For a tip loaded, wedge-shaped cantilever beam, the radial stress is identical to the three-dimensional solution. The predicted shear stress distribution in the web of a wide flange web-tapered cantilever beam subjected to a tip force and moment agrees closely with a finite element solution to the same problem. With additional verification, the elementary theory may prove to be extremely useful for the design and analysis of tapered beams under more general conditions.